
theorem Th7:
  for a, b being Real st a in ].PI/2,PI.[ & b in ].PI/2,PI.[
  holds a < b iff sin a > sin b
proof
  let a, b be Real;
  assume a in ].PI/2,PI.[ & b in ].PI/2,PI.[;
  then
A1: a in ].PI/2,PI.[ /\ dom sin & b in ].PI/2,PI.[ /\ dom sin by SIN_COS:24
,XBOOLE_0:def 4;
A2: sin a = sin.a & sin b = sin.b by SIN_COS:def 17;
  hence a < b implies sin a > sin b by A1,RFUNCT_2:21,SIN_COS2:3;
  assume
A3: sin a > sin b;
  assume a >= b;
  then a > b by A3,XXREAL_0:1;
  hence contradiction by A2,A1,A3,RFUNCT_2:21,SIN_COS2:3;
end;
